Principal Components Analysis is useful as a dimensionality reduction method. I decided to write the example below here in Python, using a combination of gists and Jupyter notebooks. For simplicity, I will describe PCA in a few steps, like in a recipe. Much of the content covered about PCA (via covariance matrix) can be found in a great paper (2005) by Jonathon Shlens. My own preference is to think about some of these approaches as flowcharts, which is what I've provided here.

Recipe for PCA (via covariance matrix)

# markdown $$ \begin{align*} & \phi(x,y) = \phi \left(\sum_{i=1}^n x_ie_i, \sum_{j=1}^n y_je_j \right) = \sum_{i=1}^n \sum_{j=1}^n x_i y_j \phi(e_i, e_j) = \\ & (x_1, \ldots, x_n) \left( \begin{array}{ccc} \phi(e_1, e_1) & \cdots & \phi(e_1, e_n) \\ \vdots & \ddots & \vdots \\ \phi(e_n, e_1) & \cdots & \phi(e_n, e_n) \end{array} \right) \left( \begin{array}{c} y_1 \\ \vdots \\ y_n \end{array} \right) \end{align*} $$
def show
@widget = Widget(params[:id])
respond_to do |format|
format.html # show.html.erb
format.json { render json: @widget }
end
end
  • Calculate deviation matrix
  • Calculate covariance matrix
  • Calculate eigenvectors and eigenvalues of the covariance matrix
  • Calculate loadings and scores

How to roll PCA from scratch

Limitations/Assumptions about PCA

PCA makes the following assumptions. There are flavors of PCA that can handle non-linear relationships, and these methods are referred to as kernel PCA.

How SVD is better

SVD can be thought of as a generalization of PCA. When we use SVD, we actually don't obtain the principal components directly, but we can easily obtain them through a few operations.

Recipe for PCA (via SVD):

  • Calculate deviation matrix
  • Perform decomposition
  • Square the diagonal matrix S, and divide by sum(S) to obtain eigenvalues
  • Matrix Vt (or U) will contain the eigenvectors

Translating from PCA to SVD